A housewife is waiting for guests and has prepared a cake. She doesn't know how many guests will come, but it will be $n-1$, $n$, or $n+1$.
What is the minimal number $f(n)$ of pieces the cake should be cut to make it possible to divide between guests equally?

$\begingroup$$f(3)=6$. To see that $f(3)\ge 6$, assume that the cake can be divided into less that 6 pieces. If 3 guests come then one of them should obtain a single piece, which means that there is a piece of size $\frac13$. But this piece is too large when 4 guests will come. So, $f(3)\le 6$. To see that $f(3)\le 6$, just divide the cake into 3 pieces of size $\frac14$ and 3 pieces of size $\frac1{12}$.$\endgroup$
– Taras BanakhMay 4 at 7:49

$\begingroup$if $n$ is odd, $3n-3$ pieces are enough, since the regular $k$-gons for $k=n-1, n, n+1$ inscribed in the same circle and sharing the same vertex have in total $3n-3$ vertices.$\endgroup$
– Fedor PetrovMay 4 at 11:11

2

$\begingroup$Maybe this is a cultural thing or something lost in translation, but it's 2019 and this is a site for professional mathematicians, so I feel it would be better if the word "housewife" were replaced by some other noun such as "mathematician".$\endgroup$
– Simon WillertonMay 7 at 20:47

3 Answers
3

Cut щге as many pieces of length $1/(n+1)+1/n+1/(n-1)$ as you can; there are $k\approx n/3$ of them. Imagine each such piece as a segment; this segment can be cut into pieces $1/(n+1),1/n,1/(n-1)$ (in this order) and $1/(n-1),1/(n+1),1/n$ (in this order). Mark cutting points for both cuttings, and cut by all of them. Notice that pieces of the same desired length do not overlap within the segment.

Thus, after $5k$ cuttings you get $2k$ non-overlapping pieces of each type separately. To arrange other $n-1-2k$ pieces of length $1/(n-1)$, take away those $2k$ pieces of length $1/(n-1)$, form a single segment of the others, and cut it into desired pieces of length $1/(n-1)$ by $n-2-2k$ cuts. Similarly, we need $(n-1-2k)+(n-2k)$ additional cuts in order to get the other two distributions possible.

$\begingroup$Sorry, but I do not understand your idea. You just cut the cake into pieces of size $\frac1{n+1}$, $\frac1{n}$ or $\frac1{n-1}$. But the pieces $\frac1n$ and $\frac1{n-1}$ are forbidden as they are too large in case $(n+1)$ guests will come.$\endgroup$
– Taras BanakhMay 4 at 22:04

$\begingroup$I think he is cutting pieces again, into subpieces, so the sizes are smaller$\endgroup$
– kodluMay 4 at 22:35

$\begingroup$@TarasBanakh: In the first part, I cut the same piece into those segments twice. Now I tried to make it clear in the text.$\endgroup$
– Ilya BogdanovMay 5 at 4:38

$\begingroup$@IlyaBogdanov Thank you for the explanations. Now I have understood. Very cute!$\endgroup$
– Taras BanakhMay 5 at 8:12

Writing down the details of the argument of Ilya Bogdanov, we can obtain the following upper bound:

Theorem.$f(n)\le\frac83n-1$ for every $n\ge 2$.

Proof. If $n=3k+1$ or $n=3k+2$, then following the idea of Ilya Bogdanov, divide the cake into $k$ pieces of length $\frac1{n-1}+\frac1n+\frac1{n-1}$. This is possible since
$k(\tfrac1n+\tfrac1{n-1}+\tfrac1{n+1})<1$. Cutting each of these pieces into 5 subpieces of lengths $$\tfrac1{n+1},\;\;\tfrac1{n-1}-\tfrac1{n+1},\;\;\tfrac1{n+1}+\tfrac1n-\tfrac1{n-1},\;\;\tfrac1{n-1}-\tfrac1n,\;\; \tfrac1n,$$ we can compose of these subpieces two pieces of any of the lengths: $\frac1{n-1}$, $\frac1n$, $\frac1{n+1}$. Cutting these $k$-pieces with 5 subpieces requires $5k+1$ cuts. To produce the remaining number of pieces it is necessary to make $((n-1)-2k-1)+(n-2k-1)+(n+1-2k-1)=3n-6k-3$ cuts. Summing up we obtain $5k+1+3n-6k-3=3n-k-2$ cuts.

Therefore, for $n=3k+1$ we have the desired upper bound:
$$
\begin{multline*}
f(n)=f(3k+1)\le 3n-k-2=(9k+3)-k-2=8k+1=\\
=\tfrac83(n-1)+1=\tfrac83n-\tfrac53<\tfrac83n-1.
\end{multline*}$$
For $n=3k+2$ we have a similar upper bound:
$$
\begin{multline*}
f(n)=f(3k+2)\le 3n-k-2=(9k+6)-k-2=8k+4=\\=\tfrac83(n-2)+4=\tfrac83n-\tfrac43<\tfrac83n-1.
\end{multline*}$$

For $n=3k$ we divide the cake into $k-1$ pieces of length $\frac1{n-1}+\frac1n+\frac1{n+1}$ and one piece of lenth $\frac1{n-1}+\frac2{n+1}$. Since
$$(k-1)(\tfrac1{n-1}+\tfrac1n+\tfrac1{n+1})+(\tfrac1{n-1}+\tfrac2{n+1})<1$$such division is possible. Then divide each of $(k-1)$ pieces like in the preceding case.
The remaining piece of length $\frac1{n-1}+\frac2{n+1}$ divide into 5 pices of lengths:
$$\tfrac1{n+1},\;\; \tfrac1{n-1}-\tfrac1{n+1},\;\;\tfrac2{n+1}-\tfrac1{n-1},\;\; \tfrac1{n-1}-\tfrac1{n+1},\;\;\tfrac1{n+1}.$$ Of these 5 subpieces we can compose either 2 pieces of length $\frac1{n-1}$ or 3 pieces of length $\frac1{n+1}$.

Then it suffices to make
$$(5k+1)+((n-1)-2k-1)+(n-2(k-1)-1)+((n+1)-(2k+1)-1)=3n-k-1$$cuts to have the required number of pieces of length $\frac1{n-1}$, $\frac1n$ or $\frac1{n+1}$.
Then
$$f(n)=f(3k)\le 3n-k-1=8k-1=\tfrac83n-1.\qquad\square$$

Remark. Comparing the known values (and upper bounds) of the function $f(n)$ for $n\le 5$ (resp. for $n\le 8$) with the upper bound $u(k)=\lfloor\frac83n-1\rfloor$, we see that $f(n)=u(n)$ only for $n=2$ and $n=4$:

$f(2)=4=u(2)$,

$f(3)=6<7=u(3)$,

$f(4)=9=u(4)$,

$f(5)=11<12=u(5)$,

$f(6)\le 14<15=u(6)$,

$f(7)=15<17=u(7)$,

$f(8)\le 19<20=u(8)$.

It is interesting to calculate the precise values of $f(n)$ for small $n\ge6$.

$f(7)\ge15$ follows from a comment of Fedor Petrov on the original question, so it suffices to find a way to cut the cake into $15$ pieces so as to serve $6$, $7$, or $8$ guests.

Let the size of the cake be $168$ (so that all the following computations involve only whole numbers). Let the $15$ pieces be of sizes $1,2,4,5,7,8,10,11,13,14,16,17,19,20,21$ (that is, every size not a multiple of $3$ up to $20$, and $21$). Then

$$1+20=2+19=4+17=5+16=7+14=8+13=10+11=21,$$

$$4+20=5+19=7+17=8+16=10+14=11+13=1+2+21(=24),$$

$$7+21=8+20=4+5+19=11+17=2+10+16=1+13+14(=28).$$

Note that this disproves my conjecture $f(n)=[5n/2]-1$ which evaluates to $16$ when $n=7$.

$\begingroup$The value $f(7)=15$ shows the difference of the problem with $n-1,n,n+1$ guests and the problem with $1,2,3,\dots,n$ guests, considered in oeis.org/A265286$\endgroup$
– Taras BanakhMay 6 at 5:48

$\begingroup$By the way, what is the exact value of $f(6)$? At the moment we have only the bounds $13\le f(6)\le 14$.$\endgroup$
– Taras BanakhMay 6 at 5:50

$\begingroup$@Taras, I'm convinced it's $14$, but every time I try to write out a proof, new cases come up that I haven't considered.$\endgroup$
– Gerry MyersonMay 6 at 6:49